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Let $q=e^{\pi i \tau}$ for $\tau \in \mathbb H$ and $$ P_2(q)=\frac{P(q)+2P(q^2)}{3}, \quad Q_2(q)=\frac{Q(q)+4Q(q^2)}{5}, \\ R_2(q)=\frac{R(q)+8R(q^2)}{9}, $$ a Fourier series of quasi-modular form or modular forms for the Fricke group $\Gamma_0^+(2)$.

In 'W. Zudilin, "The hypergeometric equation and Ramanujan functions", Ramanujan J. 7:4 (2003), 435–447' (MSN), Proposition 6 (and similarly in Proposition 7), he said that the function $P_2,Q_2,R_2$ are algebarically independent over $\mathbb C(q)$.

He mention that (table in p445) $$ P_2=3y_2+3y_1+2y_2 \\ Q_2=(y_0-y_1)(y_2-y_1) \\ R_2=(y_0-y_1)^2(y_2-y_1), $$
and Corollary 2 says that the transcendence degree over $\mathbb C$ of $\mathbb C(\tau, y_0(\tau), y_1(\tau), y_2(\tau))$ is 4. From this, we know that $P_2,Q_2,R_2$ are algebraically independent over $\mathbb C$.

But how we know over $\mathbb C(q)$?

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